65.9.13 problem 15

Internal problem ID [15779]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.1, page 186
Problem number : 15
Date solved : Thursday, October 02, 2025 at 10:27:52 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=2 \\ y^{\prime }\left (1\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 12

ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x) = 0; 
ic:=[y(1) = 2, D(y)(1) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = x \left (2-3 \ln \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 13

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==0; 
ic={y[1]==2,Derivative[1][y][1]==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x (2-3 \log (x)) \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) + y(x),0) 
ics = {y(1): 2, Subs(Derivative(y(x), x), x, 1): -1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \left (2 - 3 \log {\left (x \right )}\right ) \]